1.1 SETS

Internals of the Topic

i ROSTER
$ SET-BUILDER FORM
  • OPERATION ON SETS

CRT CONCEPT

Object : In our mathematical language, every thing in this universe ,whether living or non living is called an object.
  • Set : A set is a well defined collection of objects. The objects in a set are called its members or elements.
  • Well defined : Well defined is for a given object, it is possible to determine, whether that object belongs to the given collection or not. The following collections constitute a set :
  1. The vowels in english alphabets:
  2. All prime numbers
  3. All rivers flowing in india.
  4. The collection of all prime numbers less than20.
    Not well defined : The collection of all beautiful girls of india is not a set,since the term 'beautiful' is vague and it is not well defined.similarly 'rich persons', 'honest persons' ,'good players', 'young men' , 'yesterday', etc., do not form sets.
    I Notations : The sets are usually denoted by capital letters A, B, C, etc. The members or elements of the set are denoted by lowercase letters , etc.If is a member of the set , we write (read as belongs to
    A) and if is not a member of the set A , we write (read as x does not belong to A ). If x and y both belong to A , we write . some examples of sets used particularly in mathematics
    N : The set of all natural numbers
    or : The set of all integers
    Q : The set of all rational numbers
    R : The set of all real numbers
    : The set of all positive integers
    : The set of all positive rational numbers
    : The set of all positive real numbers
  • Representation of a Set : Usually, sets are represented in the following two ways.
  1. Roster form or Tabular form.
  2. Set Builder form or Rule Method.
Roster form : In this form, all elements of a set are listed, the elements are being separated by commas and are enclosed within brackets { } (curly brackets). For example, the set A of all odd natural numbers less than 10 in the roster form is written as
  1. In roster form, every element of the set is listed only once.
  2. The order in which the elements are listed is immaterial.
Eg 1 : Each of the following sets denotes the same set .
Eg 2 : Roster form or tabular form of set of all letters in the word 'MATHEMATICS' is given by
Note : (i) All infinite sets cannot be described in the roster form
(ii) The set of real numbers cannot be described in this form , because these elements of the set do not follow any particular pattern.
Set - Builder form : In this form, All the elements of a set possess a single common property or characterstic property which is not possessed by any element outside the set. Write a variable (say x ) representing any member of the set followed by colon( : ) or slash ( / ) which is follwed by a property satisfied by each member of the set. i.e., A set is denoted as satisfies where is the common property.
For example, the set of all prime numbers less than 10 in the set-builder form is written as is a prime number less than 10 The symbol '/' stands for the words 'such that'. Sometimes, we use the symbol ' ' in place of the symbol .
Eg : Set builder form of is is a vowel in english alphabet Classification (or) Types of Sets : Empty Set or Null Set or void set : A set which has no elements is called the null set or empty set or void set. It is denoted by the symbol or . For example, each of the following is a null set. Eg 1 : Let is a natural number} then A is the empty set because there is no natural number between land 2.
Eg 2 : The set of all real numbers whose square is -1 .
Eg 3 : The set of all rational numbers whose square is 2 .
Note : A set consisting of at element is called a non-empty
  • Singment is called a singleton set. Eg 1 : are singleton sets, which contains only one element.
    Eg 2 : Let and then , which is a singleton set.
    But and
    is not a singleton set.
    D Finite and Infinite Sets : A set which empty or consists of finite number, elements is called a finite set. Otherwise, is called an infinite set. For example, set of all days in a week is a finite set. When as, the set of all integers, denoted by . or is an integer , an infinite set.
    Cardinal Number (or) Order of a set: The number of distinct elements in a fini set is called the cardinal number of set and is denoted by or , .
    Eg : If then, :
  • Equal Sets: If A and B are two sets suc that every member of is a member of and every member of B is a member of then we say that and are equal, write as . Otherwise the sets are sai to be unequal and we write as . Eg 1: Then
    Eg 2 : A set does not change if one or mor elements of the set is repeated.
    are equi sets. That is why we generally do not reper any element in describing a set.
    Note:
Equivalent Sets : Two finite sets A and B are said to be equivalent, if . Clearly, equal sets are equivalent but equivalent sets need not be equal.
For example, the sets and are equivalent but are not equal.
Subset and Superset : Let A and B be any two sets. If every element of is an element of , then is called a subset of and we write .
If , then , B is called superset of A and we write .
  • Proper Subset : If and then is called a proper subset of and we write (read as is a proper subset of or B is a proper superset of A )
    Eg: The set Q of rational numbers is proper subset of real number set R .
    In two sets one is a subset of the other ,then the sets are called comparable sets.

- Properties of subset :

  1. Every set is a subset and a superset of itself.
  2. The empty set is the subset of every set.
  3. If A is a set with , then the number of subsets of A is and the number of proper subsets of A is .
    Note: If are any three sets, then
    i)
    ii)
    iii)
    iv)
Power Set : The set of all subsets of a given set is called power set of and is denoted by . Clearly, if A has n elements, then its power set contains exactly elements.
For example, if , then ,
.
  • Universal Set : If there are some sets under consideration, then there happens to be a set which is a superset of each one of the given sets. Such a set is known as the universal set for those sets, We shall denote by or .
    Eg 1:
    Then we consider
    as its one of the universal sets.
    Eg 2 : In the study of two dimensional geometry, the set of all points in the XYplane is called universal set.
    D Disjoint sets : If two sets A and B are such that they do not have any elements in common i.e., , then are said to be disjoint sets.
    is odd number ,
    is even number then have no common elements.
D Venn Diagram : In order to express the relationship among sets in perspective,we represent them pictorially by means of diagrams is called Venn Diagram. In these diagrams ,the universal set is represented by a rectangular region and the subsets by circles inside the rectangle. We represent disjoint sets by disjoint circles and intersecting sets by intersecting circles.

OPERATIONS ON SETS :

- Union of Two Sets :

The union of two sets and , written as (read as A union B ), is the set consisting of all the elements which are either in A or in B or in both. Thus,
or
Clearly (i) or
(ii) and
For example, if and , then .
  • Intersection of Two Sets: The intersection of two sets and , written as (read as intersection ) is the set consisting of all the common elements of A and B . Thus, and
    Clearly (i) and
    (ii) or .
For example, if and
then .
  • Difference of Two Sets :
If and are two sets, then their difference
or (or) is defined as :
and
Similarly, and
For example, if and , then and .

Important Results :

In general
  2. The sets and are disjoint sets.
  3. and
  4. and

Symmetric Difference of Two sets :

The symmetric difference of two sets and B , denoted by , is defined as


Eg: If and then
D Complement of a Set : If U is a universal set and is a subset of J the complement of A is the set which the contains those elements of , which are not contained in A and is denoted by or . Thus, and
If , . . } and , then,
Properties of complement sets
i)
ii)
iii)
iv)
v) , law of double complementation.
vi) and
are called demorgan laws

Algebra of sets :

i) Idempotent Laws : For any set A , we have
a)
b)
ii) Commutative Laws : For any two sets and , we have
a) b)
iii) Identity Laws : For any set is universal set, we have
a)
b)
c)
d)
iv) Associative Laws : For any three sets and , we have
a)
b)
v) Distributive Laws: For any three sets and , we have
a)
b) For any three sets and C
i)
ii)
For any two sets and , we have
a)
b) ,
where is the power set of .
If and are any two sets then
i)
ii)
iii)
iv)
v)
vi) ,
vii)
viii)
ix)
x)
xi)
xii)
xiii)
Properties on symmetric difference : are any three sets
i)
ii)
iii)
iv) v)
vi)
If and C are finite sets and U be the finite universal set, then
i)
ii)
iii) ,
where and are disjoint non-empty sets
iv)
v)
vi)
vii)
viii)
ix)
x) If are pair-wise disjoint sets, then
D number of elements which belong to exactly one of A or B .
  • A and B are two sets and , Then
    (i)
    (ii) ,
    (iii)
    (iv)
D No. of elements in exactly one of the sets


Do. of elements in exactly two of the sets A,B,C


Concept
Based Questions

  1. Which of the following is an empty set
  3. and
  4. The set of all even prime numbers
  1. Let and be two sets such that
    Then is equal to
  3. A
  1. Which of the following is not correct?
  1. if and only if
  2. if and only if , where is the universal set
  3. If , then
  4. is equivalent to and
  1. If and are two sets then
    is
  3. A
  1. A
  1. If and then complement of is
KEY
  1. 3
  2. 2
  3. 3
  4. 1
  5. 3

Hints Solutions

  1. .
  2. Since,
  3. satisfies (1) and (2) by definition, (4) follows trivially.
    Assuming A to be any set other than the emp set, also and , we have

    But , So (3) is incorrect
Draw venn diagram
5.

6. Complement of

公EXERCISE-II CRIO & SPO 1 피다비

ROSTER, SET-BUILDER FORM, OPERATION ON SETS

C.R.T.Q class Room Teaching Questions

  1. Which of the following not a well define collection of objects
  1. The set of Natural Numbers
  2. Rivers of India
  3. Various kinds of Triangles
  4. Five most renowned Mathematicians of th world.
  1. Write the solution set of the equation
    in roster form
  1. Write the set in set builder form
  1. where
  2. where
  3. where
  4. where
  1. Which of the following is not empty set
  1. is a natural number between 1 and 2 }
  2. and is rational
  3. is even prime number
  4. and is integer
  1. If is a letter in the word "ACCOUNTANCY" } then cardinality of A is
  1. 5
  2. 6
  3. 7
  4. 8
  1. Let be the set of all parallelograms, be the set of rectangles, be the set of rhombuses, be the set of squares and be the set of trapeziums in a plane then
  1. If the set of factors of a whole number ' n ' including ' n ' itself but not ' 1 ' is denoted by . If then ' ' is
  1. 4
  2. 8
  3. 6
  4. 10
  1. If is the set of the divisors of the number is the set of prime numbers smaller than 10 and is the set of even numbers smaller than 9, then is the set

  4. ). Let then
  1. Let and , then
  1. Let then
  1. In a class of 35 students, 24 like to play cricket and 16 like to play football also each student likes to play at least one of the two games. How many students like to play both cricket and football?
  1. 3
  2. 4
  3. 5
  4. 6
  1. In a group of 70 people, 37 like coffee, 52 like tea and each person like atleast one of the two drinks. The number of persons liking both coffee and tea is
  1. 16
  2. 13
  3. 19
  4. 20
  1. If then
  1. 6
  2. 7
  3. 8
  4. 10
  1. If and then is
  1. 50
  2. 60
  3. 70
  4. 40

S.P.R.

Student Practice Questions
16. The group of beautiful girls is
  1. a null set
  2. A finite set
  3. not a set
  4. Infinite set
  1. Which of the following is the roster form of letters of word "SCHOOL"
  1. Write the set is a positive integer and in the roaster form
  1. Which of the following is finite
  1. is set of points on a line
  2. and is prime
  3. Nand is odd
  4. and
  1. Which of the following pairs of sets are equal
  1. is letter of word "ALLOY"
    is letter of word "LOYAL"
  3. and
  1. Let then the number of subsets of is
  1. 2
  2. 4
  3. 8
  4. 0
  1. How many elements has , if
  1. 1
  2. 2
  3. 3
  4. 0
  1. then
  1. If , then
  1. Let then
  1. Let then
  4. A
  1. In a committee 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. The number of persons speaking at least one of these two languages is
  1. 60
  2. 40
  3. 38
  1. If and are two sets such that
  1. 58 and , then is equal to
  2. 240
  3. 50
  4. 40
  5. 20
  2. or



  5. we have,


  8. Beautiful is a relative term but not well defined.
  9. elements not repeated and denoted by small letters.
  10. all are <40
  13. no of subsets
  15. All the elements in and .

公EXERCISE-II CRITO & SPO LEVELLI

ROSTER, SET-BUILDER FORM, OPERATION ON SETS

C.R.T.Q Gass Room Teaching Questions

  1. Let . If represent any member of , then roster form of but
  1. Two finite sets have and elements. If total number of subsets of the first set is more than that of the total number of subsets of the second. The values of and respectively are
  1. 7,6
  2. 6,3
    3)5,1
  3. 8,7
  1. If and then
  1. If then the power set of is
  1. A
  1. The smallest set such that is
  1. If sets and are defined as

    , then
  1. If then
  2. 21 N
  3. 10 N
  4. 4 N
  5. 5 N
  6. If , then
  1. A
  1. If , , then is equal to
  1. 400
  2. 240
  3. 300
  4. 500
  1. If and , then is
  1. 27
  2. 28
  3. 29
  4. 30
  1. If , then the value of is
  1. 18
  2. 15
  3. 16
  4. 17
  1. Let and . If then ' ' is
  1. 4
  2. 5
  3. 6
  4. 7

S.P.P.

Student Practice Questions

  1. Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and foot ball, and 12 play foot ball and cricket. Eight play all the three games. The total number of members in the three athletic teams is
  1. 43
  2. 76
  3. 49
  4. 53
  1. If sets and have 3 and 6 elements each, then the minimum number of elements in is
  1. 3
  2. 6
  3. 9
  4. 18
  1. If then greatest value of and value of are
  1. 60,43
  2. 50,36
  3. 70,44
  4. 60,38
  1. If is a multiple of 4 and a multiple of then consists of multiples of
  1. 16
  2. 12
  3. 8
  4. 4
  1. Two finite sets have and elements. Th total number of subsets of the first se is 48 more than the total number subsets of the second set. The values and are
  1. 7,6
  2. 7,6
  3. 6,4
  4. 7,4
  1. If and then
  1. Let be the set of non-negative integer is the set of integers, is the set non-positive integers, is the set of ever integers and is the set of prim numbers then.
  1. In a class of students, student have passed in Mathematices and students have passed in physics. students who have passed in physic only is
  1. 22
  2. 33
  3. 10
  1. 90 students take mathematics, 72 tak science in a class of 120 students. If 1 take neither Mathematics nor scienc then number of students who take botl the subjects is
  1. 52
  2. 110
  3. 162
  4. 100
  1. A set has 3 elements and another se has 6 elements. Then
  1. In a survey of 400 students in a schoo 100 were listed as taking apple juice, 15 as taking orange juice and 75 were liste
    as taking both apple as well as orange juice. then how many students were taking neither apple juice nor orange juice are
  1. 120
  2. 220
  3. 225
  4. 150

KEV

  1. 3
  2. 2
  3. 1
  4. 3
  5. 2
  6. 3
  7. 1
  8. 3
  9. 3
  10. 1
  11. 2
  12. 2
  13. 1
  14. 2
  15. 1
  16. 2
  17. 3
  18. 2
  19. 4
  20. 4
  21. 1
  22. 3
  23. 3

Hints
\section*{Solutions}

  1. but




  3. is a multiple of 49 for all
    A contains elements which are multiple of 49 and clearly B contains all multiples of 49.
  4. no of subsets
  5. Since {given }
    atleast
  6. The graph of and do not intersect

  10. ,


  11. .



  12. ,
    ,




  13. ,

    Total
    no.of players
Greatest value of
Least value of
16. .
.
is a multiple of 12
17.
18. Consider common part
19.
  1. ,
  1. and then

三 EXERCISE-III 京 CRTO & SPO LISILII

OPERATION ON SETS

C.R.T.Q

Class Room Taaching Questions

  1. Let and be two sets then
  4. U
  1. A set contains elements. The number of subsets of this set containing more than elements is equal to
  1. From 50 students taking examin in Mathematics, Physics and each of the students has passed in one of the subject, Mathematics, 24 Phys 19 Chemistry. At mossis, at Mathematics and Ph Chemistry Mathematics and most 20 Physics and Chemistry. the largest possible number that, have passed all three examinations
  1. 16
  2. 14
  3. 18
  1. Which is the simplified represents of where are subsets of set
  1. A
  2. B
  3. C
  1. The set is equal to
  1. If and then is
  1. Suppose are thirty sets ei with five elements and are ni each with three elements such th . If each element of belongs to exactly ten of the 's and easy 9 of the , then the value of is
  1. 15
  2. 135
  3. 45
If and , where are relatively prime, then
  4. none

S.P.Q. Student Practice Questions

. A survey show that in a city that of the citizens like tea where as like coffee. If like both tea and coffee, then
  1. An investigator interviewed 100 students to determine their preferences for the three drinks: milk (M), coffee(C) and tea (T). He reported the following : 10 students had all the three drinks 20 had and only: 30 had and had M and had M only; 5 had C only; 8 had T only. Then how many did not take any of the three drinks
  1. 20
  2. 3
  3. 36
  4. 42
    l. In a college of 300 students, every students reads 5 newspapers and every newspaper is read by 60 students. The number of newspapers is
  5. atleast 30
  6. atmost 20
  7. exactly 25
  8. atmost10
  1. In a class of 55 students the numbers of students studying different subjects are 23 in mathematics, 24 in physics, 19 in chemistry, 12 in mathematics and physics, 9 in mathematics and chemistry 7 in physics and chemistry and 4 in all the three subjects. the numbers of students who have taken exactly one subject is
  1. 6
  2. 13
  3. 16
  4. 22
  1. Out of 800 boys in a school. 224 played cricket 240 played hockey and 336
    played basketball. of the total, 64 played both bas-ketball and hockey, 80 played cricket and basketball and 40 played cricket and hoc-key, 24 played all the three games. The num-bers of boys who did not play any game is
  1. 128
  2. 216
  3. 240
  4. 160
  1. In a certian town families own a phone and own a car, families own neither a phone nor a car. 2000 families own both a car and a phone. Consider the following statements in this regard.
    i. families 0 w both a car and a phone
    ii. 35% families own either a car or a phone
    iii. 40,000 families live in the town.
Which of the above statements are correct?
  1. i and ii
  2. i and iii
  3. ii and iii
  4. i,ii and iii
  1. In a battle of the combatants lost one eye, an ear, an arm, a leg, lost all the four limbs the minimum value of is
  1. 10
  2. 12
  3. 15
  4. 5

KEY

  1. 1
  2. 4
  3. 2
  4. 3
  5. 1
  6. 2
  7. 3
  8. 1
  9. 4
  10. 1
  11. 3
  12. 4
  13. 4
  14. 3
  15. 1

Hints Solutions

が a
＝d゙ （d゙ up） U
A゙ A゙ A゙
2．Let the original set contains elements，then subsets of this set containing more than in elements means subsets containing elements， elements ．．．． elements．
Required number of subsets
.
3．The given conditions can be expressed as ， and ．

Therefore，the number of students is at most
4． draw venn diagram．
5．



6． either
or ．
7．
Again，
Thus
8．We have the hey positive integral multiples of ， the set of positive inter multiples of C ．
the set of positive integ multiples of be be and relatively prime｜Hence， be
9．Let the population of the city be loo Let A denote the set of citizens who tea and denote the set of citizen， like coffee．
and
，



Also and

and

From（1）and（2）：

10． ．The numbers can be read fi the fig，number of people who did not any drink



11. If n is the required number of newspapers then
12. ,
,
.
We have to find ,
,

Now





.





,




13. .
,
,





.
14.

Since
, Now


.
Thus .
But
of the total
Total numbers of families
.
Since and total number
of families and .
(ii) and (iii) are correct.
15. Minimum value of
.

三XXERCISE-IV LEVELL-IV

Assertion - Reason Type :

Note :
  1. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
  2. Statement-1 is true, Statement-2 is true, Statement-2 is not correct explanation for Statement-1.
    3)Statement-1 is true, Statement-2 is false.
    4)Statement- 1 is false, Statement- 2 is false.
  1. Statement-1: If is a prime number and then is a prime number .
    Statement-2: If then .
  2. Statement-1: If , then .
Statement-2: and .
3. Statement-1: then

Statement-2: .
4. Statement-1: If then number of elements in power set of
Statement-2: If then number of elements in power set of
5. Statement-1: If is wh. number is natural , then is whole number
Statement-2: If then
6. Statement-1: If and are dissoint sets then
Statement-2: If and are sets then
7. Statement-1: If then numbe of proper subsets of
Statement-2: If then number of proper subsets of ,
8. Match the following sets for all
a)
b) A
c) B
d)
i)
ii)
iii)
iv)
  1. i-b, ii-c, iii-a, iv-d
  2. i-b, ii-c, iii-d, iv
  3. i-b, ii-c, iii-c, iv-d
  4. i-d, ii-c, iii-a, i.

KEV

  1. 1
  2. 4
  3. 2
  4. 3
  5. 1
  6. 1
  7. 2
  8. 1

Hints Solutions

  1. The set of Prime numbers are the subs of the Natural number set.
  2. and .




  3. t.
  4. The set of natural numbers are the sub set of the whole numbers set
  5. For disjoint sets no common elements
  6. Number of proper sub sets
  7. Draw venn diagram

INTEGER & NUMERICAL

ANSWER TYPE QUESTIONS

  1. Let .
If is a multiple of 2 and is a multiple of 7 , then the number of elements in the smallest subset of X containing both A and B is
  1. Let is a 3-digit number and for some . If the sum of all the elements of the set is , then is equal to .
    b. If and
    , then the value of expression is_
  2. A function is given by then the sum of the series

    is equal to :

Answers

  1. 29.00
  2. 5
  3. 2
  4. 4

Hints Solutions




  2. B and C will contain three digit numbers of the form and respectively. We need to find sum of all elements in the set effectively.
    Now, where denotes sum of elements of set .
Also, 992}
Case-I: If then

which is not possible as given sum is
.
Case-II : If then


  1. replace by
    1. Some important results on cardinal numbers :